ON THE INTEGRATION 

 OF CERTAIN DIFFERENTIAL EQUATIONS*. 



No. II. 



IN the last number of the Journal f, a method was given for 

 the investigation of a class of differential equations, by means of 

 successive reductions. 



The present communication contains solutions of some ana- 

 logous equations effected by a similar process. The results will 

 however be exhibited in a very different form. 



We begin by taking a particular case of the equations in 

 question, 



(I), 



x 

 where p is an integer. 



Let y = S<ve tt , 



.*. n(n l)...(w m + 2) (n m + l pm) a n + ka n _ m = 0...(2). 



Assume 

 a n = {n m + 1 (p 1) m] {n m+ 1 (p 2) m] 



...(n-m + l)/(fc)J. ...... (3), 



f(k) being some function of Jc, to be determined hereafter. 

 Then 



{n m + 1 (p 1) m} 



...... (4). 



* Cambridge Mathematical Journal, No. XI. Vol. n. p. 193, February, 1841. 

 t Page 96 of this volume. 



